Section A — Chapters 1–22

Constructions I

Junior Cycle — 1st Year

✓Identify and correctly use a compass and a straight edge for geometric constructions.
✓Construct the perpendicular bisector of a given line segment.
✓Construct the bisector of a given angle.
✓Construct a line perpendicular to a given line, passing through a point on the line or a point not on the line.
✓Construct a line parallel to a given line, passing through a given point.
✓Construct an angle of 60 degrees without using a protractor.
✓Construct a triangle given the lengths of its three sides.

Key concepts

Construction Equipment

In geometry, 'construction' means drawing shapes, lines, and angles accurately using only a compass and a straight edge (a ruler without markings for measurement). A sharp pencil is also essential for accuracy. We do not use a protractor for geometric constructions, only for measuring angles after construction.

Construction 1: Perpendicular Bisector of a Line Segment

The perpendicular bisector of a line segment is a line that cuts the segment into two equal parts and is at right angles (90°) to it. Every point on the perpendicular bisector is equidistant from the endpoints of the segment.

Construction 2: Bisector of an Angle

The bisector of an angle is a ray that divides the angle into two angles of equal measure. Every point on the angle bisector is equidistant from the two arms of the angle.

Construction 4: Perpendicular to a Line from a Point on the Line

This construction creates a line that forms a 90° angle with a given line, passing through a specific point that lies on the given line.

Construction 5: Perpendicular to a Line from a Point Not on the Line

This construction creates a line that forms a 90° angle with a given line, passing through a specific point that does not lie on the given line.

Construction 6: Line Parallel to a Given Line Through a Given Point

Parallel lines are lines in a plane that are always the same distance apart and never meet. This construction creates a new line that is parallel to an existing line and passes through a specified point. One common method uses the property of corresponding angles.

Construction 8: Angle of 60 Degrees

An angle of 60 degrees can be constructed accurately using only a compass and a straight edge, without the need for a protractor. This construction forms the basis of an equilateral triangle, where all angles are 60 degrees.

Construction 9: Triangle Given Three Sides (SSS)

To construct a triangle when the lengths of all three sides are known (Side-Side-Side or SSS), we use a compass to mark off the side lengths from a base line. This method ensures the triangle has the exact specified dimensions.

Key facts to remember

1Geometric constructions in Junior Cycle maths use only a compass and a straight edge (ruler without markings).
2A perpendicular bisector divides a line segment into two equal parts and forms a 90° angle with it.
3An angle bisector divides an angle into two equal angles.
4Perpendicular lines meet at a right angle (90°).
5Parallel lines are always the same distance apart and never intersect.
6An angle of 60° is the internal angle of an equilateral triangle.
7Construction lines (arcs) must be left visible as part of your answer in an exam.

Worked examples

Example 1

a) Construct the perpendicular bisector of the line segment AB, where A and B are two points. b) Construct the bisector of the angle PQR.

Ia) Perpendicular Bisector of AB:

II1. Place the compass point on A. Open the compass to more than half the length of AB. Draw an arc above and below the line segment.

III2. Without changing the compass width, place the compass point on B. Draw another arc above and below the line segment, intersecting the first two arcs.

IV3. Use a straight edge to draw a straight line connecting the two points where the arcs intersect. This line is the perpendicular bisector of AB.

Vb) Bisector of Angle PQR:

VI1. Place the compass point on Q (the vertex of the angle). Draw an arc that intersects both arms of the angle (QP and QR). Label the intersection points C and D.

VII2. Place the compass point on C. Draw an arc in the interior of the angle.

VIII3. Without changing the compass width, place the compass point on D. Draw another arc in the interior of the angle, intersecting the previous arc. Label the intersection point E.

94. Use a straight edge to draw a straight line from Q through E. This line is the bisector of angle PQR.

Answer

The constructions are complete as described in the steps. The perpendicular bisector of AB is drawn, and the bisector of angle PQR is drawn.

Ensure your compass width is consistent for intersecting arcs to maintain accuracy.

Example 2

a) Construct a line perpendicular to line L at point P on L. b) Construct a line parallel to line M, passing through point Q (not on M).

Ia) Perpendicular to L at P:

II1. Draw a line L and mark a point P on it. Place the compass point on P. Draw arcs of equal radius on both sides of P along line L. Label these points X and Y.

III2. Place the compass point on X. Open the compass to a radius greater than XP. Draw an arc above P.

IV3. Without changing the compass width, place the compass point on Y. Draw another arc above P, intersecting the first arc. Label the intersection point Z.

V4. Use a straight edge to draw a straight line from P through Z. This line is perpendicular to L at P.

VIb) Parallel to M through Q:

VII1. Draw a line M and mark a point Q not on M. Draw a transversal line from Q that intersects line M at any point, say R. This creates an angle at R.

VIII2. Place the compass point on R. Draw an arc that intersects both the line M and the transversal line QR. Label the intersection points S and T.

93. Without changing the compass width, place the compass point on Q. Draw a similar arc that intersects the transversal line QR. Label the intersection point U.

104. Measure the distance between S and T with your compass. Place the compass point on U. Draw an arc that intersects the arc drawn from Q. Label the intersection point V.

115. Use a straight edge to draw a straight line from Q through V. This line is parallel to M.

Answer

The constructions are complete. A line perpendicular to L at P is drawn, and a line parallel to M through Q is drawn.

There are multiple methods for constructing parallel lines; this method uses the property of corresponding angles.

Example 3

a) Construct an angle of 60 degrees at point A on a given line. b) Construct a triangle with sides of length 5 cm, 6 cm, and 7 cm.

Ia) Angle of 60 degrees at A:

II1. Draw a line and mark a point A on it. This will be one arm of the angle.

III2. Place the compass point on A. Draw an arc that intersects the line. Label the intersection point B.

IV3. Without changing the compass width, place the compass point on B. Draw another arc that intersects the first arc. Label the intersection point C.

V4. Use a straight edge to draw a straight line from A through C. The angle CAB is 60 degrees.

VIb) Triangle with sides 5 cm, 6 cm, 7 cm:

VII1. Draw a line segment of 7 cm. Label its endpoints D and E.

VIII2. Place the compass point on D. Open the compass to 5 cm. Draw an arc above the line segment DE.

93. Place the compass point on E. Open the compass to 6 cm. Draw another arc above the line segment DE, intersecting the first arc. Label the intersection point F.

104. Use a straight edge to draw straight lines connecting D to F and E to F. Triangle DEF is the required triangle.

Answer

The constructions are complete. An angle of 60 degrees is drawn, and a triangle with sides 5 cm, 6 cm, and 7 cm is constructed.

Always start with the longest side as the base for triangle constructions to ensure arcs intersect easily.

Common mistakes

✗Using a ruler to measure lengths or angles instead of constructing them with a compass and straight edge.
✗Not using a sharp pencil, leading to thick, inaccurate lines and arcs that make precise intersection points difficult to identify.
✗Changing the compass width accidentally during a construction step, which compromises accuracy.
✗Drawing arcs that are too small or do not intersect clearly, making it hard to find the exact intersection points.
✗Not labelling points or lines, making the construction difficult to follow and understand for an examiner.

Exam tips

★Practice each construction multiple times to become proficient and confident in your technique.
★Always use a sharp pencil and a good quality compass and straight edge for maximum accuracy.
★Leave all construction lines (arcs) visible; they are an essential part of your solution and demonstrate your method to the examiner.
★Be precise with your compass placements and the drawing of lines to ensure accurate results, as marks are often awarded for accuracy.

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